Lower bound

A lower bound of a subset $A$ of a partially ordered set $(P,\le)$ is an element $\ell\in P$ such that $\ell\le a$ for all $a\in A$. The subset $A$ is bounded below if it has at least one lower bound in $P$.

Lower bounds complement upper bounds and lead to greatest lower bounds (the infimum in real analysis).

Examples:

• In $(\mathbb{Z},\le)$, the integer $-5$ is a lower bound for the subset $\{-2,0,3\}$.
• In the poset $(\mathcal P(X),\subseteq)$, the set $A\cap B$ is a lower bound for $\{A,B\}$, where $A\cap B$ is the intersection of $A$ and $B$.